Introduction to Theoretical Physics/Mathematical Foundations/Arithmetic, Physics, and Mathematics
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Tentative outline for Arithmetic, Physics, and Mathematics; or the Units of Measurement
Contents
Units of Measurement as Mathematical Constants[edit]
 Physics and Mathematics begin with counting
 1 apple, 2 apples, etc.
 Thus an "apple" in our records is a unit of measurement, the quantity in question being "number of apples"
 This evolves into simple arithmetic
 1 apple added to 1 apple is 2 apples
 10 apples subtracted from 30 apples is 20 apples
 Introduction of shorthand notation
 Mathematics can discard the physical objects in question and has the luxury of concerning itself with abstract concepts
 Whereas in mathematics the constant represents a numerical constant, in physics this constant can represent a physical constant, thereby allowing physical objects to behave as mathematical entities in mathematical equations
 Units of measurements are important in mathematical equations since they represent critical information and errors in calculations will result if these physical constants are neglected
is wrong in the sense that
is the only answer allowed under the rules of mathematics
 Also, care must be taken when we perform mathematical operations
represents 9 apples arranged in a square
creates a new physical quantity apple(orange) that is neither apples nor oranges! This is how new physical quantities, i.e. energy are created from lengths, times, and masses.
Basic Units of Measurement[edit]
 Time
 Usually measured in seconds
 Shorthand is s
 10 seconds
 10 s
 Shorthand is s
 Only unit of measurement not to be decimalized (although such a system does exist)
 Usually measured in seconds
 Distance
 Usually measured in meters
 Shorthand is m
 10 meters
 10 m
 Shorthand is m
 Usually measured in meters
 Mass
 Base unit is the kilogram
 Shorthand is kg
 10 kilograms
 10 kg
 Shorthand is kg
 Sometimes measured in grams
 Shorthand is g
 10 grams
 10 g
 Shorthand is g
 Base unit is the kilogram
Derived Units of Measurement[edit]
 Area
 Usually measured in meters squared
 Usually measured in meters squared
 Volume
 Usually measured in meters cubed
 Usually measured in meters cubed
 Density
 Linear density
 Usually measured in kilograms per meter
 Usually measured in kilograms per meter
 Area density
 Usually measured in kilograms per meter squared
 Usually measured in kilograms per meter squared
 Volumetric density
 Usually measured in kilograms per meters cubed
 Usually measured in kilograms per meters cubed
 Linear density
Scientific Notation[edit]
 Shorthand notation for large or tiny numbers based on powers of 10
 Large
 Small
Système International d'Unités (International System of Units, aka SI)[edit]
 Further simplification of written numbers


The Mathematics of Conversion Between Units[edit]
 In mathematical equations, units of measurement behave as constants
 To convert from one unit of to another, we utilize an equation relating the two measurements
 We can solve and substitute for the constant
The Mathematics of Conversion Between Units
1. In mathematical equations, units of measurement behave as constants * (1\mbox{ m} + 2\mbox{ m})\times 4\mbox{ m} = 12\mbox{ m}^2 2. To convert from one unit of to another, we utilize an equation relating the two measurements * 1\mbox{ km} = 1000\mbox{ m} \, 3. We can solve and substitute for the constant m * \frac{1}{1000}\mbox{ km} = \mbox{ m} * \left[1\left(\frac{1}{1000}\mbox{ km}\right) + 2\left(\frac{1}{1000}\mbox{ km}\right)\right]\times 4\left(\frac{1}{1000}\mbox{ km}\right) = 12 \left(\frac{1}{1000}\mbox{ km}\right)^2 * \left(1\times 10^{3}\mbox{ km} + 2\times 10^{3}\mbox{ km}\right)\times 4\times 10^{3}\mbox{ km} = 12\times 10^{6}\mbox{ km}^2
A Physicists' View of Calculus[edit]
 The derivative and small quantities
 The integral and summation of infinite quantities